Does there exist such a measure for an infinite-dimensional $H$? There is surely no such Gaussian measure, for example.
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zhorasterNov 15 '10 at 18:15

Good point, Mark. I've edited the question to include the assumption that $H$ is infinite-dimensional.
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Tom LaGattaNov 15 '10 at 18:29

zhoraster, there does exist such a measure. This is Corollary 2 in Section III.2.2 of <i>Probability Distributions on Banach Spaces</i> by Vakhania, Tarieladze and Chobanyan. Furthermore, there does exist a Gaussian measure with covariance operator $I$. This is typical in the abstract Wiener space formalism for constructing Gaussian measures on Banach spaces.
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Tom LaGattaNov 15 '10 at 18:30

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In general $|\langle \cdot, h\rangle|$ will not be in the closure of $H^*$. Maybe you want to ask whether the polynomials in elements of $H^*$ are dense?
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Bill JohnsonNov 15 '10 at 18:48